If you are just looking for a solution, you sure would use Primes built-in methods ("root" function or a solve block)
But I guess this is an exercise in programming and implementing an algorithm in Prime.
You also want the whole series of iteration steps as the result.
I spotted a couple of errors in your sheet:
1) The guess values 0;0 result in a non-invertible Jacobi-Matrix and so your function must fail
2) You defined f as a function with two arguments (x and y) but in your Newton function you used the function argument f as a function with just one vector argument. You should rewrite your input function f to accept a vector as its argument and do the very same with your function J. That way you may also be able to deal with more than two variables, using the very same function "Newton". You may just redefine J(x):=J(x[1,x[2) to save typing the definition from anew.x
At the top you are collecting the current variable values and the number of the iteration step in a matrix "result"
3) You forgot about the second matrix index (should bi "i") in result
4) When writing x_i you used the literal index instead of the vector index (Thats what the error message is trying to tell you)
Furthermore "result" in the for-loop is accidentally labelled as "function" - it must be manually re-labelled as "Variable"
5) You completely missed the correct iteration step using J(x), etc. You simply calculated the new x-value being the result of f(x).
If you correct those errors and you provide meaningful guess values, your routine should work OK (I renamed you function f to F, so I can use the previously defined function f (and g) to check the result using Primes root-function.
